Periods of Oscillation of Pendulums

Purpose:
The purpose of the experiment was to improve the ability to derive moments of inertia, and then use those moments of inertia to predict the period of oscillation as if the object were a pendulum.

Equipment Used:

The object, either a half circle or an isosceles triangle, is hung from a pivot and its period of oscillation is recorded by the photogate which records the time it takes for one oscillation to occur.











Data Collected:
First, the half sphere was oscillated, both from its base and at the apex of its curve. The only data that is needed to calculate both of the periods of oscillation for this experiment is the radius of the half circle and the location of its center of mass. The mass isn't important since the equation for torque and angular acceleration ensure that the oscillation is independent of the mass when it is acting like a pendulum.
Second, the isosceles triangle was oscillated both from the center of its base and from the tip where both equal sides come together. The only data that is needed to calculate both of the periods of oscillation for this experiment is the base and height of the triangle and the location of the center of mass.
Half Circle Data:
Radius = .125m
Center of mass = 4R/3pi
Theoretical period of oscillation from base = .77s
Experimental period of oscillation from base = .768s
Theoretical period of oscillation from apex = .755s
Experimental period of oscillation from apex = .764s

Triangle Data:
Base = .078m
Height = .145m
Center of mass = H/3
Theoretical period of oscillation from tip = .670s
Experimental period of oscillation from tip = .676s
Theoretical period of oscillation from base = .560s
Experimental period of oscillation from base = .565s

Calculations:

Half circle calculations:

These calculations are for deriving the moment of inertia of the half circle as if the axis of rotation. The moment of inertia turns out to be .5MR^2

These calculations are for deriving the theoretical period of oscillation, which is .77s. The second portion of the calculations are for deriving the moment of inertia if the axis of rotation was at the apex of the curve. It is done by using the parallel axis theorem twice, once to move the axis of rotation to the center of mass from the base of the half circle, then again to move the axis of rotation to the apex of the circle. The parallel axis theorem only works for when the axis of rotation is at the center of mass.

These calculations are for deriving the theoretical period of oscillation at the new location, which is .7548s

















Triangle calculations:

These calculations are for deriving the moment of inertia of the triangle if the axis of rotation was at the tip of the triangle. The moment of inertia was M((1/24)B^2 + (1/2)H^2).

These calculations are for the period of oscillation for the triangle if oscillated from the tip. The second portion of the calculations are for calculating the moment of inertia if the axis of rotation was from the base of the triangle. The final portion of the calculations are for calculating the theoretical period of oscillation which is .5596s














Conclusion:
For both the half circle and the triangle, the predictions were very close to experimental value, and that is because there was only a single variable to calculate for. The only reason that the prediction didn't match the actual period was because sin(angle) is approximated as being only the angle, in radians.

Sincerely,

Swaggy C